Stefan-Boltzmann Law

The Stefan-Boltzmann law describes the power radiated from a black body
radiator. The Watts radiated = s (T14 -
T24), where s = (2p5 k4 )/(15c2 h3) 
a constant, and T1
is the (black body) temperature (in K) and T2 is the ambient temperature. For
this exercise, T1 was 3100K, and T2 was 293K (20C)
at 12W radiated power. The
first term represents radiation of the black body to ambient. The second term represents
radiation of the ambient to the black body. (T2 is so small as to have
essentially no effect for this discussion. It is included for completeness) A tungsten filament may be approximated
by a black body.

Given the above equation for 12W radiated, and T1 = 3100K,
then s can be determined. Given this, one can
determine T1 for other radiated powers. In this exercise, 9W
yields a T1 of 2884K, and 6W yields a T1 of 2607K.

Efficacy (lumens/watt) measures the visual efficiency of light -- how
well we see it. The efficacy changes dramatically for filaments of varying
temperature. Shown here is a curve of efficacy versus ideal radiator temperature.
Note that the highest practical filament temperature is about 3200K. The sun
radiates at 5785K - -near the peak of the curve. A filament radiating at
3100K is about 22 LpW. A filament radiating at 2600K is only about 8 LpW.